1. Field of the Invention
The present invention relates to a determination method of determining a light intensity distribution (effective light source) to be formed on the pupil plane of an illumination optical system, an exposure method, and a storage medium.
2. Description of the Related Art
An exposure apparatus is employed to fabricate a semiconductor device using a photolithography technique. The exposure apparatus projects and transfers the pattern of a mask (reticle) onto a substrate (for example, a wafer) by a projection optical system. To keep up with the recent advances in micropatterning of semiconductor devices, the exposure apparatus requires a technique of attaining a high resolution.
Because an exposure apparatus cannot always ensure an ideal amount of exposure on the substrate and an ideal focus position, it may transfer a pattern different from that having a desired shape (mask pattern shape) onto a substrate. The amount of exposure deviates from an ideal state due to factors such as instability of a light source and nonuniformity of the illuminance distribution in an illumination region. Also, the focus position deviates from an ideal state due to factors such as instability of the holding position of the substrate and unevenness of the substrate. A model defined by the ranges of amounts of exposure and focus positions, within which a desired pattern can be transferred onto the substrate, is called a process window, and the exposure apparatus requires a technique of attaining a wide process window.
Oblique-incidence illumination, for example, is known as a technique for attaining both a high resolution and a wide process window. In the oblique-incidence illumination, a mask is obliquely irradiated with exposure light using an annular effective light source (the light intensity distribution on the pupil plane of an illumination optical system) or an effective light source with a shape having a plurality of (for example, two or four) poles. The annular effective light source is defined by two degrees of freedom (parameters): the annular zone radius and the annular zone width. Thus, the following technique has been proposed. Pattern images for effective light sources defined by those two degrees of freedom are obtained by simulation while changing them to various values, and an annular zone radius and an annular zone width are selected based on these pattern images, thereby determining an optimum effective light source.
Also, in recent years, T. Matsuyama, et. al., “A Study of Source & Mask Optimization for ArF Scanners”, Proc. of SPIE, USA, SPIE, 2009, Vol. 7,274, p. 727,408 (literature 1), proposes a technique which increases the number of degrees of freedom which define the effective light source. In the technique described in literature 1, the pupil plane of an illumination optical system is divided into a plurality of regions in a grid pattern, and light intensities are individually set for the respective divided regions. However, assuming, for example, that the pupil plane of the illumination optical system is divided into 63×63 regions, a thousand or more degrees of freedom are determined. From the viewpoint of the computation time, it is not realistic to obtain pattern images for respective combinations of degrees of freedom defined within such a wide optimization space to determine an optimum effective light source. Although Japanese Patent No. 3342631 proposes a heuristic optimization technique of adjusting the initial value and iterating computation to obtain an optimum solution, this technique may not only require a long computation time but also result in a local solution.
On the other hand, Japanese Patent No. 4378266 proposes a technique which uses mathematical programming in effective light source optimization with such large degrees of freedom. The mathematical programming mathematically guarantees its solution to be optimum, and can shorten the computation time.
The technique described in Japanese Patent No. 4378266 is designed to apply approximation to a maximization problem for the process window to transform this problem into one type of mathematical programming, that is, a linear programming problem to be solved, thereby obtaining a solution. The process window is generally the product of the range of amounts of exposure and that of focus positions, within which the width of a pattern image falls within a tolerance. In the technique described in Japanese Patent No. 4378266, the positions of the two side edges of a line pattern image are defined, and the effective light source is optimized under this condition.
Unfortunately, the technique described in Japanese Patent No. 4378266 often cannot optimize the effective light source. A case in which, for example, a mask pattern is formed by equidistantly arranging three line patterns having the same size will be considered. In this case, when attention is focused on the central line pattern, the position (midpoint position) of the central line pattern and that of an image corresponding to this central line pattern coincide with each other unless asymmetric aberrations such as coma and distortion are present. On the other hand, when attention is focused on the line pattern at each end, a difference occurs between the position of the line pattern at each end and that of an image corresponding to this line pattern at each end due to an optical proximity effect even if no asymmetric aberrations are present. The difference between the position of a pattern and that of an image corresponding to this pattern is called a pattern shift, which generally occurs for most patterns. In the technique described in Japanese Patent No. 4378266, the position coordinates of the two edges of the image of the line pattern are fixed, and an effective light source is obtained such that the two edges of the image of the line pattern are positioned at these position coordinates, so this technique does not take the concept of a pattern shift into consideration. Therefore, when a pattern shift occurs, the technique described in Japanese Patent No. 4378266 cannot be used to determine an optimum effective light source.
Also, with advances in micropatterning of semiconductor devices, the occurrence of a pattern shift may have a more serious influence on the yield (throughput). A mask for an ion implantation process in an SRAM will be considered as an example. This mask has a simple line-and-space pattern, which has a relatively large pitch of several hundred nanometers. In a mask of this type, a demand for the line width of its pattern is not so strict, but the overlay accuracy on an isolation oxide layer present in the underlying layer is of prime importance in an ion implantation process, so a demand for a pattern shift is strict. Therefore, the effective light source must be optimized (determined) such that the amount of pattern shift for the mask pattern falls within a tolerance.